Sets, Relations & Functions PYQs Set 7 (PDF)

Set 7 of 25 · Questions 181-210 of 734

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 7 (Q181-210)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
181. $The derivative of a^{sec x} w.r.t. a^{tan x} (a > 0) is$ (1) $sec x a^{sec x - tan x}$ (2) $sin x a^{tan x - sec x}$ (3) $sin x a^{sec x - tan x}$ (4) $a^{sec x - tan x}$ 182. $lim n \to \infty \frac{n !}{( n + 1 ) ! - n !} =$ (1) $1$ (2) $-1$ (3) $2$ (4) $0$ 183. $If x 1 + y + y 1 + x = 0, then \frac{dy}{dx} =$ (1) $\frac{- 1}{( 1 + x ) ^{2}}$ (2) $\frac{1}{( 1 + x ) ^{2}}$ (3) $\frac{2}{( 1 + x ) ^{3/2}}$ (4) $\frac{- 2}{( 1 + x ) ^{1/2}}$ 184. $If sin x = \frac{2 t}{1 + t ^{2}}, tan y = \frac{2 t}{1 - t ^{2}}, then \frac{dy}{dx} is equal to$ (1) $1$ (2) $-1$ (3) $2$ (4) $0$ 185. $Let A = {x : x \in R; x is not a positive integer} Define f : A \to R as f (x) = \frac{2 x}{x - 1}, then f is$ (1) $injective but not surjective$ (2) $surjective but not injective$ (3) $bijective$ (4) $neither injective nor surjective$ 186. $f (x) = \frac{x}{e ^{x} - 1} + \frac{x}{2} + 2 cos^{3} \frac{x}{2} on R - 0 is$ (1) $one one function$ (2) $bijection$ (3) $algebraic function$ (4) $even function$ 187. $An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?$ (1) $15$ (2) $21$ (3) $10$ (4) $9$ 188. $lim x \to 0 \frac{x tan 4 x - 2 x tan 2 x}{( 1 - cos 4 x ) ^{2}} =$ (1) $\frac{1}{8}$ (2) $\frac{1}{4}$ (3) $\frac{1}{2}$ (4) $1$ 189. $The function f (x) = cot x is discontintous on every point of the set$ (1) $(x = (2 n + 1) \frac{π}{2}; n \in Z)$ (2) ${x = nπ; n \in Z}$ (3) $(x = \frac{nπ}{2}; n \in Z)$ (4) ${x = 2 nπ; n \in Z}$ 190. $If f (0) = 0, f (1) = 1, f (2) = 2 and f (x) = f (x - 2) + f (x - 3) for x = 3, 4, 5 \dots, then f (10) =$ (1) $13$ (2) $9$ (3) $11$ (4) $10$ 191. $Consider the following relation R on the set of real square matrices of order 3. R = {(A, B)\|A = P-1 BP for some invertible matrix P}. Statement -1 : R is equivalence relation. Statement - 2 : For any two invertible 3 \times 3 matrices M and N, (MN)-1 = N-1M-1.$ (1) $Statement-1 is true, statement-2 is a correct explanation for statement-1.$ (2) $Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for statement-1.$ (3) $Statement-1 is true, statement-2 is false.$ (4) $Statement-1 is false, statement-2 is true.$ 192. $Let A = x \in R : \| x+1 \| <2 and B = {x \in R : \| x - 1 \| \geq 2} . Then which one of the following statements is NOT true?$ (1) $A - B = (- 1, 1)$ (2) $B - A = R - (- 3, 1)$ (3) $A \cap B = (- 3, - 1]$ (4) $A \cup B = R - [1, 3)$ 193. $If x = cosec θ - sin θ, y = cosec^{2022} θ - sin^{2022} θ and (\frac{dy}{dx})^{2} = \frac{k ( y ^{2} + 4 )}{g ( x )} where k \in R, then 10 + k - g (2022) =$ (1) $0$ (2) $6$ (3) $10$ (4) $14$ 194. $Let f : R \to R be defined by f (x) = 2 x + 6 which is a bijective mapping then f^{- 1} (x) is given by$ (1) $\frac{x}{2} - 3$ (2) $2 x + 6$ (3) $x - 3$ (4) $6 x + 2$ 195. $If g (x) = \frac{x}{[ x ]} for x > 2, then lim_{x \to 2} \frac{g ( x ) - g ( 2 )}{x - 2} is equal to$ (1) $-1$ (2) $0$ (3) $\frac{1}{2}$ (4) $1$ 196. $Let f : R - 0 \to (- \infty, 1) be a polynomial of degree 2, satisfying f (x) f (\frac{1}{x}) = f (x) + f (\frac{1}{x}) . If f (K) = - 2 K, then the sum of squares of all possible values of K is:$ (1) $7$ (2) $6$ (3) $1$ (4) $9$ 197. $If y = sin x \cdot sin 2 x \cdot sin 3 x \dots ... \cdot sin nx, then y^{’} is$ (1) $\sumk=1n ktankx$ (2) $y\cdot\sumk=1n kcotkx$ (3) $y\cdot\sumk=1n ktankn$ (4) $\sum_{k = 1}^{n} cot kx$ 198. $Let \sum_{k = 1}^{10} f (a + k) = 16 (2^{10} - 1), where the function f satisfies f (x + y) = f (x) f (y) for all natural numbers x, y and f (1) = 2 . Then the natural number a is :$ (1) $2$ (2) $16$ (3) $4$ (4) $3$ 199. $Let A = -3, -2, -1, 0, 1, 2, 3 and R be a relation on A defined by xRy if and only if 2 x - y \in 0, 1 . Let l be the number of elements in R . Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then l + m + n is equal to:$ (1) $17$ (2) $18$ (3) $15$ (4) $16$ 200. $The function f (x) = [x], where [x] denotes the greatest integer not greater than x, is$ (1) $continuous only at positive integral values of x$ (2) $continuous for all non-integral values of x$ (3) $continuous only at rational values of x$ (4) $continuous for all real values of x$ 201. $Let R = (1, 2), (2, 3), (3, 3) be a relation defined on the set 1, 2, 3, 4 . Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:$ (1) $10$ (2) $7$ (3) $8$ (4) $9$ 202. $The sum of all the elements in the range of f (x) = Sgn (sinx) + Sgn (cosx) + Sgn (tanx) + Sgn (cotx), x \neq = \frac{nπ}{2}, n \in Z, where Sgn (t) = (1, if t > 0 - 1, if t < 0), is :$ (1) $2$ (2) $0$ (3) $4$ (4) $-2$ 203. $If f (x) = log_{x^{2}} (log_{e} x), then f^{’} (x) at x = e is$ (1) $1/2 e$ (2) $0$ (3) $1$ (4) $1/ e$ 204. $If a set A has m elements and set B has n elements and the number of injections from A to B is 2520. Then, m is equal to$ (1) $2$ (2) $7$ (3) $6$ (4) $5$ 205. $Suppose that the number of elements in set A is p, the number of elements in set B is q and the number of elements in A \times B is 7 then p^{2} + q^{2} =$ (1) $50$ (2) $42$ (3) $51$ (4) $49$ 206. $Let A = -2, -1, 0, 1, 2, 3 . Let R be a relation on A defined by xRy if and only if y = max x, 1 . Let l be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then l + m + n is equal to$ (1) $11$ (2) $12$ (3) $14$ (4) $13$ 207. $If y = x + x + x + \dots . \infty, then \frac{dy}{dx} =$ (1) $\frac{1}{y ^{2} - 1}$ (2) $\frac{1}{2 y + 1}$ (3) $\frac{2 y}{y ^{2} - 1}$ (4) $\frac{1}{2 y - 1}$ 208. $If , where denotes the greatest integer less than or equal to and represents the fractional part of , then is equal to_______$ 209. $The number of symmetric relations defined on the set 1, 2, 3, 4 which are not reflexive is______$ 210. $Let a, b and c be three non-zero non-coplanar vectors. Let the position vectors of four points A, B, C and D be a - b + c, λa - 3 b + 4 c, - a + 2 b - 3 c and 2 a - 4 b + 6 c respectively. If AB, AC and AD are coplanar, then λ is$